JPH0215384A - Interpolation and digital differential analysis - Google Patents

Abstract

PURPOSE: To easily execute a digital differential analysis(DDA) with a simple constitution by performing the DDA by using a hardware multiplier accumulator. CONSTITUTION: The output registers 410 to 414 of a multiplier accumulator are initialized via a CPU, etc. As for the incremental change value of the output value over the line of a prescribed length to be interpolated in a first input register 102, the value which is equal to the reciprocal of the line length is loaded into a second input register 104. The contents of the registers 102 and 104 is multiplied by a multiplication accumulator 408, the cumulative results of the addition of the accumulation of a fraction part, the overflow part and the integer part of the multiplication result and an overflow part via an adder 408 are stored in the registers 410 to 414, respectively, and the interpolation by the DDA is performed. By the execution of the DDA for which this hardware is used, a constitution is simplified and the execution of the DDA can be exactly performed without performing a division operation, etc.

Description

DETAILED DESCRIPTION OF THE INVENTION [Field of Industrial Application] The present invention relates to interpolation using a multiplier accumulator (MAC) and digital differential analysis.
This relates to a method of analysis.

[Prior Art] i) Importance of Interpolation In the field of computers, interpolating values is
often required. For example, to draw a line on a bitmapped display monitor, the desired line may have to be drawn between several pixels displayed between the start and end points. In that case, the line is written by lighting up the pixels that form the line.

Another example is that for a given area of the displayed image,
This is a case where you want to gradually add shadows. The conventional method for this operation is to determine the starting and ending shades of the shadow above a given pixel.
ing 5hade), the shadow value (shasdin
gvalue).

These methods are by no means tedious. It also really only scratches the surface of possible applications for interpolation algorithms.

-Numerical problems are as follows.

Over a given number of steps (N), the initial value (V
) gradually changes by a certain m (DALTAV).

One conventional method for performing interpolation is the use of a digital differential analyzer (hereinafter referred to as DDA). 1
One well-known example is DDA for drawing lines.

A typical simple line DDA divides the X, X elements of the line into an integer part and a fractional part! I'm going to have a good time. Then, to DD, increase the X and X elements in fractional units. When the fractional part is full, the integer part is incremented. D above
DA uses integer elements to form its output. As used in the computer field, one of the elements increases steadily by one, while the other increases conditionally.

ii) Simple I) Eight Simple The DA algorithm will be better understood with reference to FIG. For example, assume that a line 102 is drawn between a starting point 104 and an ending point 106 on a display monitor. And the points are where the grid lines intersect (
Assume that you can only plot (corresponding to pixels on your monitor). The initial value (v) of the line is zero (
(starting point value of the X element). Expected changes (DALTA
V) is 5 (the end point value of the X element), and its value changes (
N) The number of steps to do is 7 (total length of the line in the direction of the X axis)
It is.

The simple basic language DDA shown in Example 1-1 below is used to make such plots.

Example 1-1 DDA program X-0:Y-0:N-7:DELTA V-5Y-V+
, 5 FOR1-I To N PLOT(TRUNC(X),'TRUNC(Y)Y-
Y+DELTAV/N X-X+1 EXTX Pl, OT (TRtlNC(X), TRUNC(
Y) ND The following is understood from the above example. [The IDA program simply initializes the Y value to the Y start coordinate value of the line plus the rounding factor.
r) Set to 0.5. Then, in N iteration cycles, plot the X and Y values at points in integer elements and Y!
Increase the NI coordinate by the slope of the line (DELT
8V/to), increases the X element of the line by 1. The last plot command is for plotting the last point of the line.

Goraud shading is a program similar to the above program.
There are some things used in The difference between the two programs is
The difference between changing the X-axis and Y-axis coordinates is that the shading value changes over a given number of steps. Similarly, the shading program (shadiB program)
Now, we don't need the last plot command.

1ii) Breseham's algorithm
nham'sAIgorithm) A simple [IDA operation (operation
The problem with ) is that the division operation
ration). This includes CP
Traditionally, simple DDA was considered unsuitable for hardware implementation because it is too expensive in terms of U-time. One of the traditional common methods to avoid splitting
Another type of interpolation method is known as Presenham's algorithm.

At each iteration, Presenham's algorithm
It is designed to add 1 or minus 1 to the coordinate value. Whether the remaining coordinate values are incremented depends on the error term maintained by the algorithm (i.e.
According to the value of Presenham's error term). This error term is used to remember the distance between the exact path of a theoretically complete line and the actual pixel that was addressed and written (i.e. shining as a dot). An example of Presenham's algorithmic lineblotting program, and other examples of ODA, can be found in Principles of Ractive Computer Graphics (2nd Edition) by William, M., Newman, Robert, F., Sproul, and McGraw-Hill. ).

This document is incorporated herein by reference.

Although Presenham's algorithm solves the partitioning problem, it suffers from several other problems, as do many conventional DD8 techniques.

iv) The problem of non-fractional slopes Many conventionally known DD8 hexagramming techniques (including Presenham's algorithm program) only allow the interpolated value to increment by one with restrictions. There is. This problem is more clearly understood when considering lines with slopes greater than 45 degrees from the X-axis. In these cases, N (number of steps) is always smaller than DELD^V (intended change in value).

The problem with this situation is illustrated in FIG.

Figure 2 shows a line 202 drawn from the starting point at the origin of the X and Y axes, and an ending point 2 at the X and Y positions m4.7.
06 is shown. In this case, N is equal to 4 (Sunawachi, 4
-7-(f-s0'), DELTA V is equal to 7 (i.e., 7 minus 0).

Those skilled in the art will understand that under such circumstances, in Presenham's algorithm routine, X and y will always increase. The result is a 45 degree plot, as indicated by the dot in FIG. This is, of course, not the expected result.

To adapt to such conditions, DELTA in the Y direction
Another code must be added to interpolate the value of X over V steps. In other words, if the slope direction is less than or equal to 1, the program must be rewritten to accommodate it. Depending on the angle of the line, Presenham's algorithm or other conventional D
Programs using the DA program must enter one set of codes or another. Programmers and programs must keep track of which sections are being executed.

Unfortunately, although the above method solves the line drawing problem,
Gouraudshad
-dimensional problems such as ing) are not solved. for example,
Two shades represented by 60 and 155 (5a
Consider interpolating between de). variable DELT
A V is equal to 195 (255-Go) and N is 8
be equivalent to. X is initially equal to 1 and Y is initially equal to 6o.

The result of dividing DELTA V by N is 7.5, and Presenham's algorithm-type program divides each pixel (i.e., 60, 8L62, 83, 84, 65, 66
．． 67), we only increase each shade value across the stars. Reversing the slope in this case would give meaningless results. This can be easily understood by considering that N and X are defined in pixels, whereas Y and DELTA V are defined in shading codes.

■) Line clipping (Line C1C11p
pin and subset line (Subset Li
ne) Problem Another problem with Presenham's algorithm and many other interpolation methods is said to be the subset line problem. Simply put, in two-dimensional sketch graphics, it is difficult to determine the correct plotting position of a line when erasing or repeating the line. For example, assume that a line Ll is drawn between a starting point pixel PI and an ending point pixel P2 on the screen. The chosen interpolation algorithm shall then draw the line by plotting the pixel groups Gl (eg, by brightening the first color). If it is necessary to erase or overwrite part of the line, it will be necessary to retraverse part of the line L (for example, by shining a second color).

For this operation, a new line L2 must be drawn by selecting a starting pixel P4 and an ending pixel P5 within the first pixel group P4. Since the new line L2 is used to erase a portion of line Ll, it is important that the group of pixels plotted when drawing line L2 (referred to as group G2) is a subset of G1. .

The problem of subset lines is not a trivial problem. If the above conditions are not met, the line cannot be completely erased and extra pixels will be plotted, thus cluttering the screen. It is understood that the general Presenham algorithm does not satisfy the above conditions. Some suggestions are provided by the authors to circumvent this problem. One example recommends the use of a look-up table that performs the digitization of possible lines on the screen stored therein. Such suggestions are generally considered unrealistic. The problem of subset lines is described in detail in the book ``Data Structures for Raster Graphics''.
, edited by Line Lierock, P., and published by Spring Arbor Log. The content thereof shall be referred to in its entirety as a cited document in this specification.

This tree is from the Eurographics Seminars Series (
Eurographics Sem1nars 5er
ies).

Despite the preliminary results of this tree regarding the subset line problem, this is still a problem that must be solved.

I don't even know if a good algorithm exists. Of course, that's because it's complicated. (Bage 7, Baragraph 1).

The issue of subset lines has become important.

In particular, when cutting a line at the border of a displayed window, its location is important. This problem is illustrated in FIG. If a line displayed on a screen or window is computed from a point outside the screen or window, the line must be clipped.

In FIG. 3, line 302 starts at point 304 outside window 306. In FIG. A portion of line 302 must be displayed in the window while the remaining portion is hidden. In order to solve this problem, two methods have been conventionally used.

One method of clipping is to calculate the starting point of the unclipped supline and perform a Presenham-type interpolation. In this case, the interpolation starts from a point other than the actual origin of the line and encounters the problem of subset lines.

This results in the clipped line containing slightly different pixels than those drawn in the original line. If it is necessary to follow a line that is not clipped, or if the line is clipped in another window, the programmer will find that a slightly different set of pixels has been selected.

A second conventional method circumvents the problem of subset lines by performing Plessenham-type interpolation from the start of the line, tracking the point where the window starts, but attempting to solve the problem by not writing to non-window areas. It is something to do. This method is correct. However, it requires extra software expense and computation time to ensure that lines are not written to non-window areas. Moreover, they spend their time doing all the ODA even for invisible Bixel.

Clipping and interpolation techniques, DDA and bowloud shading, including Presenham's algorithm, are discussed in William M. Newman and Robert F. Subroull (10th ed. 19
84) in “°Interactive Computer Graphics” (2nd edition).

vi) Programmer's Dilemma Programmers traditionally face a dilemma. The programmer must choose between the simpler and DA-type methods. In addition to requiring expensive partitioning, that method creates a subset line problem and
Presenham's algorithm or similar technique is needed to limit the size of the increments to 1.

An interpolation method that can interpolate the entire range of values over any number of steps is desirable. Over any range of values or any number of steps, DDA-type methods can be used to
It is desirable to be able to perform a shading operation. It would be desirable to be able to increase the interpolated value by a number greater than 1 at each step.

It is desirable to avoid software partitioning. On top of that,
For lines clipped at separate clipping boundaries, it is particularly desirable to be able to perform fast and accurate line clipping to ensure accurate repetition of selected pixels.

SUMMARY OF THE INVENTION The present invention includes a method for performing digital differential analysis using a hardware multiplier accumulator (MAC).

[Example] Hereinafter, the present invention will be explained in detail based on Examples.

The present invention utilizes a hardware multiplier accumulator (MA
G::ll1ultiplier accumulat
or) to perform digital differential analysis. This method allows programming when the number of incremental steps is smaller than the range of values in which the steps exist. This method solves some inherent problems that existed previously in clipping methods using Presenham's algorithm and other conventional 1) DA techniques.

The basic hardware according to the preferred embodiment of the present invention is
As shown in Fig. and Fig. 5. 4 and 5 show block diagrams of a multiplier accumulator (MAC).

The MAC 400 has two input registers 402, 404,
It includes a multiplier array 406, an adder 408, and three output registers 410, 412, 414. 1.5P (
least 51gn1ficantproduct)
The register stores and accumulates the fractional part of the multiplication result. M
SP (most 51gn1fic ant prod
uct) register is used to store and accumulate the integer part of the result. XTP (extended prod
uct) register is used to store and accumulate overflows.

The MAC 400 has output registers 410, 412, 414
Preferably, it has a function of preloading and can operate in at least multiplication mode and multiplication/accumulation mode.

A preferred multiplier accumulator (MAC) is the IDT72
1016x 16 parallel multiplier accumulator available from Integr/Ited Device Technology, Santa Monica, California. TDT72
The structure and operation of 10 are based on the company's “old GHPER FORMA
NCE CMO5DATA HΔND[1OOK”.

FIG. 5 shows another embodiment of the invention. In this example, 64
kXI69 - Hardware that includes a read-only memory (ROM) lookup table 502. Lookup table 502 is used to find the reciprocal of all values appearing at its address input.

Next, the operation of this embodiment will be explained with reference to FIG. First, the MAC's output register is preloaded first.

-Let's consider drawing the lines of a tree. The LSP register is first loaded with an initial rounding value (eg, 0.5). One coordinate i+h (for example, Y Itll )
The initial value of V is initially loaded into the MSP. M.A.G.
must be set to multiply-accumulate mode when the input is loaded.

Next, DELT8V is stored by the address input of the first input register 402. At the same time, the value of N is
It is stored at the address input of lookup table 502 and held until l/N appears and stabilizes at first input register 402.404. They are input to the first and second input registers 402 and 404, respectively, in synchronization with the clock.

The value of DELTA V is also clocked into the first input register as soon as it is stable at the MAG input. Then, when the reciprocal of the N value becomes stable, it is input to the second input register in synchronization with the clock. Data stability at the M to G input is easily ensured by employing an appropriate locking mechanism that takes into account ROM access time.

Since it is set to multiply-accumulate mode, M
AC multiplies 1/N by DEL DEV. This result is added to the initial contents of output registers 410, 412, 414. The accumulated value is input to the output register 410.412.41.4 in synchronization with the clock. next,
The result of the multiplication is reclocked and accumulated with the output register value. This process continues to repeat for the total number of steps for which interpolation is to be done.

This simply means that the MAC output registers 410, 412
414 a number of times equal to the number of steps in which the DELTA V value is to be changed.

In order to correctly plot the values of X, Y, X is incremented by 1 each clock cycle while the value of Y to be plotted is stored in the MSP register 412 each clock cycle. When Y has an initial value other than zero, the MSP register 412 is preloaded to that value (YO) during initialization. The ability to load initial Y values has great significance in line clipping operations.

When line clipping is performed, the MAC is programmed in the same way as for line generation. In the initial multiply-accumulate operation, the first input register 402 is loaded with the number of steps to reach the visible portion of the line (i.e., n*DELTA
V), the woody initial settings are the same.

The first multiply-accumulate operation is as if n steps were taken to achieve this configuration, including a 16-bit fractional error term (i.e., the result is n◆DELTAV/N ), set up the output registers. Thus, the first point is plotted.

The first input register is then loaded with DELTA V as a standard line trowing routine. The algorithm then proceeds until the end of the line is normally reached or the line intersects the window. MAC is used for the initial calculation of n*DELTA V.

In a preferred embodiment of the method of the invention, N and DELTA
V is expressed as a 16-bit binary number. ROM lookup table 502 is programmed with the reciprocal of each address in each address area, preferably in unsigned fractional representation.

In other words, 2 must have the hexadecimal value 8000. Address 4 must have 4000. Addresses 1 and 0 are special cases, and the MAC programming algorithm (see Figure 5) must consider the zero case. The M to C operation is not required when plotting only a single point. This is because it can be fully processed by software.

Conveniently, negative slope values are not required. In Golaut shading, DDA always starts from the numerically lowest shade and works its way up. When drawing a line, DDA always proceeds from a point where the end point results in a positive slope. - Once programmed, the output of the look-up table Rf)M naturally yields the reciprocal of the N value held at the address input.

Conveniently, each input to the MAC is on a separate bus, so the two inputs can be loaded together or separately. Similarly, output registers are clocked separately from input registers. Therefore, to draw a line or shade a line (5hades), the input register only needs to be clocked - degrees, and to clip to a given region, the input register only needs to be clocked twice. .

Conclusion Many improvements can be made by those skilled in the art by practicing the present invention. For example, larger (i.e.
(16 bits or more) MAC can be used to improve accuracy. A multiplexer can be placed on the output side to select which part of the output register is used. This allows different mixes of integer and fractional sizes for different applications.

The system and method according to the invention can be modified to use the adder only in the inner calculation loop to calculate NIDELTAV once with the accumulator at the beginning of the algorithm. This is particularly useful if a high degree of accuracy is required.

The method according to the invention can be easily modified using the same hardware to vary the number of configured steps of the ODA. This is very useful when using parallel processors.

That is, it is useful when there is more than one processor computing the shading of pixels along a line. For example, if a graphics system has 5 processors in the X direction (each responsible for one pixel in a group of 5), then 5 shading values need to be calculated in -steps. The system and method of the present invention facilitates this by setting up each processor's MAC to perform five steps every cycle.

Therefore, the preferred embodiment described above has been described as a typical example of the present invention, and changes and modifications can be made without departing from the gist of the present invention.

[Effects of the Invention] According to the method of the present invention, the programmer can obtain the best method of linear interpolation. MAC hardware is often used to handle cases where the number of coarsely increasing steps is smaller than the range of values found for the steps, without the partitioning inherent in simple DDA, and without the subset line problems associated with Presenham's algorithm. DDA can be executed quickly without requiring any additional code required.

By performing DDA using hardware,
Many problems with linear interpolation are solved. For example, in one embodiment, line clipping can be done easily and accurately. By properly initializing the MAC, lines of any pitch will be quickly clipped at the window borders. By adding a hardware lookup table to one of the MAC inputs, there is no need to partition the operation of the system, thus eliminating the need for a Presenham-type algorithm. This system and method solves a long unsolved problem with subset lines.

Moreover, the tree method can be used to interpolate any range of values or any number of steps. The relative sizes of N and DELT8V do not affect this method in any way. The present invention eliminates the need for extra code. The present invention also allows ODA to be performed to solve one-dimensional problems where the range of values is larger than the number of steps across the chips on which the interpolation is performed.

In some embodiments, the method includes a bow-loud line
Used for shading. One of the MMC inputs is loaded with the delta color (the amount of color varies along the line) and the other input is loaded with the inverse of the line length. The high part of the M to G result register (MSC) is preloaded with the starter color. By using the ItOM lookup table, the reciprocal of the line length does not need to be calculated programmatically. The program simply knows the actual line length when entering the lookup table. The reciprocal is then automatically generated and no software calculations are required. Whenever a pixel is drawn, the two inputs of the MAC are multiplied and added to the first result. The high part of the result is used as the color of each pixel. Then, when the accumulated lower part of the result overflows, it changes automatically.

With the method of the invention, each pixel change is a delta color divided by the length of the line. Therefore, the last color is the first color plus delta color.

Conveniently, this method is compatible with many CPUs implementing bit slice processors and microprocessors.
It is possible to use simple MAC hardware that is already available. MAC integrated circuits can be inexpensively implemented into systems that do not already use them.

In one embodiment of the method according to the invention, a look-up table can be used to perform automatic reciprocal operations when data is provided for processing by the MAC. Using this lookup table eliminates the need for time-consuming divide operations, reducing CPU costs.

Therefore, the method of the present invention does not require algorithms that perform partitioning operations as previously required.

The method of the invention can be used to solve many interpolation problems. And in many cases it costs nothing.

[Brief explanation of the drawing]

Figure 1 shows a typical line interpolation problem, Figure 2 shows the
FIG. 3 is a diagram illustrating a problem with clipping; FIG. 4 is a block diagram illustrating a multiplier accumulator according to an embodiment of the present invention; FIG. FIG. 6 is a block diagram showing a multiplier accumulator and lookup table memory according to another embodiment of the tree invention, and FIG. 6 is a flowchart showing the ODA processing procedure according to one embodiment of the invention. Y-axis Figure 1 Y-axis Figure 2

Claims (1)

[Claims] 1) A method of performing linear interpolation over a line of predetermined length, comprising the steps of: (a) initializing any of the output registers of a multiplier accumulator with a predetermined initial value; ) loading a first input register of the multiplier accumulator with a value equal to the incremental change in output value over the length of the line; and (c) loading a second input register of the multiplier accumulator with a value equal to the incremental change in output value over the length of the line. (d) multiplying the values of its first and second inputs by said multiplier accumulator; and (e) loading the fractional part of the result of step (d) to a first output. (f) adding an integer portion of the result of step (d) with the overflow of the first output register and accumulating it in a second output register to form an accumulated result. A linear interpolation method characterized by 2) to generate Gouraud shading, the predetermined value used to initialize the second output register represents a starting shade, and the incremental change represents a change in shade over the length of the line; , wherein the accumulated result represents a shade value for a predetermined display area. 3. The method of claim 1, further comprising the step of: 3) determining the reciprocal of the line length using a lookup table stored in memory. 4) Loading the output of the lookup table directly into the second input register.
The method described in. 3. The method of claim 2, further comprising the step of: 5) repeating steps (d), (e) and (f) a predetermined number of times equal to the number of display areas in which the color change is to occur. Method described. 6) The method of claim 5, wherein each of the display areas is a pixel. 7) The method of claim 5, wherein each of the display areas is a patch. 8) A method for displaying only a predetermined segment of the line by clipping a line having two directional components and a predetermined length, the method comprising: (b) loading a first input register of the multiplier accumulator with a value equal to the length of one directional component of the line; (c) loading a second input register of said multiplier accumulator with a value equal to the reciprocal of the length of the other directional component of said line length; (e) accumulating a fractional portion of the result of step (d) in a first output register; and (f) overflowing the first output register in a second output register. (g) accumulating an integer portion of the result of step (d) in said second output register; and (h) displaying a point on a display device each time the value in the second output register changes. the pixel having at least one coordinate value determined in response to a value stored in the second output register. 9) The method of claim 8, wherein the predetermined value used to initialize the second output register represents the starting point of a predetermined segment of the line. 9. The method of claim 8, further comprising: 10) determining the reciprocal of the line length using a lookup table stored in memory. 11. The method of claim 10, further comprising: 11) directly loading the output of the lookup table into the second input register. 12) Steps (d), (e), (f), (g)
12. A method according to any one of claims 8 to 11, further comprising repeating (h) a predetermined number of times equal to the number of points at which the change in line length is to occur. 13) The method of claim 12, wherein each of the points is a pixel. 14) The method of claim 12, wherein each of the points is a patch.